Civil Engineering Reference
In-Depth Information
V
V
V
V
0
13 −−− +
−++−
0
838
83813
0
y
y
y
y
27
EI
L
0
1
2
3
1
−
=
3
y
y
y
y
0
1
2
3
2
3
0
54 76
25 60
0
V
V
V
V
0
.
.
1
kips
=
2
3
The exact solution at the end is
V
0
=
75
.
00
k
, which is a huge error, while
the value in the center is exact. The large error at the end is due to the fact
that the shear and moment drop dramatically at that point, which creates a
discontinuity in the equation.
The final example of using difference operators to solve differential equa-
tions is the critical buckling load of a column. The critical buckling load
of a pinned end column is sometimes included in strength of materials, but
will be derived here. The derivations start with the differential equation of
the elastic curve similar to a beam. The deflected column under the critical
load is shown in Figure 3.8.
dy
dx
2
M
EI
2
= ′′ =−
y
At any point
x
along the column, there is a deflection
y
that will produce an
eccentric moment in the column equal to
P
cr
y.
This is used in the equation
of the elastic curve as follows:
Z
y
P
cr
P
cr
X
A
B
X
L
Figure 3.8.
Numeric modeling with difference operators.
